## Numerical approximation for x+\ln(1+e^{-x})

One other mystery from the numerical solution is the function that handles the evaluation of

$x+\ln(1+e^{-x})$

when the argument is extreme.  I also notated this onto paper, so it can be evaluated, I think it has to do with the precision of a float in c.  Perhaps it can be improved for a double in C#.